The $n\text{th}$ partial sum of the series $\sum\limits_{n=1}^{\infty }{{{a}_{n}}}$ is given by ${{S}_{n}}=\frac{3n+2}{2n}$. $a_6=$
${{a}_{6}}={{S}_{6}}-{{S}_{5}}$ ${{a}_{6}}=\frac{20}{12}-\frac{17}{10}=\frac{5}{3}-\frac{17}{10}=-\frac{1}{30}$